algebra 8 9 study guide the quadratic formula and...

Algebra I 8-9 Study Guide Page ! of !1 21

The Quadratic Formula and the Discriminant Attendance Problems. Evaluate x = - 2, y = 3, and z = -1.

1. ! 2. xyz 3. ! 4. y - xz

5. -x 6. !

• I can solve quadratic equations by using the Quadratic Formula. • I can determine the number of solutions of a quadratic equation by using the

discriminant.

Vocabulary: Discriminant

Question: What do a call a person who is crazy about solving quadratic equations? Answer: A quadratic fanatic.

"The road to success is always under construction.”—Actress and Comedian, Lilly Tomlin

x2 x2 − yz

z2 − xy


Common Core CCSS.MATH.CONTENT.HSA.REI.B.4 Solve quadratic equations in one variable.

CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

CCSS.MATH.CONTENT.HSA.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

You have completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula. The Quadratic Formula is the only method that can be used to solve any quadratic equation.


Quadratic Equation Song (Pop Goes the Weasel)

!

X equals negative B plus or minus the square root Of B squared minus four AC, All over two A!

x = −b ± b2 − 4ac2a


1E X A M P L E Using the Quadratic Formula

Solve using the Quadratic Formula.

A 2 x 2 + 3x - 5 = 02 x 2 + 3x + (-5) = 0 Identify a, b, and c.

Use the Quadratic Formula.

Substitute 2 for a, 3 for b, and -5 for c.

Simplify.

Simplify.

Write as two equations.

Simplify.

Write in standard form. Identify a, b, and c.

Substitute 1 for a, -2 for b, and -3 for c.

Simplify.

Simplify.


Simplify.

x = -b ± √ """" b 2 - 4ac __ 2a

x = -3 ±

√ """""" 3 2 - 4 (2) (-5)

__ 2 (2)

x = -3 ±

√ """" 9 - (-40)

__ 4

x = -3 ± √ " 49 _

4 = -3 ± 7 _

4

x = -3 + 7 _ 4

or x = -3 - 7 _ 4

x = 1 or x = - 5 _ 2

B 2x = x 2 - 31 x 2 + (-2) x + (-3) = 0

x = - (-2) ±

√ """"""" (-2) 2 - 4 (1) (-3)

___ 2 (1)

x = 2 ±

√ """" 4 - (-12)

__ 2

x = 2 ± √ " 16 _

2 = 2 ± 4 _

2

x = 2 + 4 _ 2

or x = 2 - 4 _ 2

x = 3 or x = -1

Solve using the Quadratic Formula. 1a. -3 x 2 + 5x + 2 = 0 1b. 2 - 5 x 2 = -9x

Many quadratic equations can be solved by graphing, factoring, taking the square root, or completing the square. Some cannot be easily solved by any of these methods, but you can use the Quadratic Formula to solve any quadratic equation.

2E X A M P L E Using the Quadratic Formula to Estimate Solutions

Solve x 2 - 2x - 4 = 0 using the Quadratic Formula.

x = - (-2) ± √ """"""" (-2) 2 - 4 (1) (-4)

___ 2 (1)

x = 2 ±

√ """" 4 - (-16)

__ 2

= 2 ± √ " 20 _

2

x = 2 + √ " 20 _

2 or x = 2 -

√ " 20 _ 2

Use a calculator: x ≈ 3.24 or x ≈ -1.24.

2. Solve 2 x 2 - 8x + 1 = 0 using the Quadratic Formula.

You can graph the related quadratic function to see if your solutions are reasonable.

Check reasonableness

8-9 The Quadratic Formula and the Discriminant 583

CS10_A1_MESE612225_C08L09.indd 583CS10_A1_MESE612225_C08L09.indd 583 2/10/11 5:14:24 AM2/10/11 5:14:24 AM


Helpful Hint


Example 1. Solve using the Quadratic Formula. A. ! B. ! 6x2 + 5x − 4 = 0 x2 = x + 20


Guided Practice. Solve using the Quadratic Formula. 7. ! 8. ! −3x2 + 5x + 2 = 0 2 − 5x2 = −9x


Many quadratic equations can be solved by graphing, factoring, taking the square root, or completing the square. Some cannot be solved by any of these methods, but you can always use the Quadratic Formula to solve any quadratic equation.

1E X A M P L E Using the Quadratic Formula

Solve using the Quadratic Formula.

A 2 x 2 + 3x - 5 = 02 x 2 + 3x + (-5) = 0 Identify a, b, and c.

Use the Quadratic Formula.

Substitute 2 for a, 3 for b, and -5 for c.

Simplify.

Simplify.


Simplify.

Write in standard form. Identify a, b, and c.

Substitute 1 for a, -2 for b, and -3 for c.

Simplify.

Simplify.


Simplify.

x = -b ± √ """" b 2 - 4ac __ 2a

x = -3 ±

√ """""" 3 2 - 4 (2) (-5)

__ 2 (2)

x = -3 ±

√ """" 9 - (-40)

__ 4

x = -3 ± √ " 49 _

4 = -3 ± 7 _

4

x = -3 + 7 _ 4

or x = -3 - 7 _ 4

x = 1 or x = - 5 _ 2

B 2x = x 2 - 31 x 2 + (-2) x + (-3) = 0

x = - (-2) ±

√ """"""" (-2) 2 - 4 (1) (-3)

___ 2 (1)

x = 2 ±

√ """" 4 - (-12)

__ 2

x = 2 ± √ " 16 _

2 = 2 ± 4 _

2

x = 2 + 4 _ 2

or x = 2 - 4 _ 2

x = 3 or x = -1

Solve using the Quadratic Formula. 1a. -3 x 2 + 5x + 2 = 0 1b. 2 - 5 x 2 = -9x

Many quadratic equations can be solved by graphing, factoring, taking the square root, or completing the square. Some cannot be easily solved by any of these methods, but you can use the Quadratic Formula to solve any quadratic equation.

2E X A M P L E Using the Quadratic Formula to Estimate Solutions

Solve x 2 - 2x - 4 = 0 using the Quadratic Formula.

x = - (-2) ± √ """"""" (-2) 2 - 4 (1) (-4)

___ 2 (1)

x = 2 ±

√ """" 4 - (-16)

__ 2

= 2 ± √ " 20 _

2

x = 2 + √ " 20 _

2 or x = 2 -

√ " 20 _ 2

Use a calculator: x ≈ 3.24 or x ≈ -1.24.

2. Solve 2 x 2 - 8x + 1 = 0 using the Quadratic Formula.


Check reasonableness




Example 2. Solve ! by using the quadratic formula. x2 + 3x− 7 = 0


9. Guided Practice. Solve ! using the quadratic formula. 2x2 − 8x+1= 0


If the quadratic equation is in standard form, the discriminant of a quadratic equation is b2 – 4ac, the part of the equation under the radical sign. Recall that quadratic equations can have two, one, or no real solutions. You can determine the number of solutions of a quadratic equation by evaluating its discriminant.

Desmos: The Discriminant of a Quadratic Function (FZ4N)

If the quadratic equation is in standard form, the discriminant of a quadratic equation is b 2 - 4ac, the part of the equation under the radical sign. Recall that quadratic equations can have two, one, or no real solutions. You can determine the number of solutions of a quadratic equation by evaluating its discriminant.

Equation x 2 - 4x + 3 = 0 x 2 + 2x + 1 = 0 x 2 - 2x + 2 = 0

Discriminant a = 1, b = -4, c = 3

b 2 - 4ac

(-4) 2 - 4 (1) (3)

16 - 12

4

The discriminantis positive.

a = 1, b = 2, c = 1

b 2 - 4ac

2 2 - 4 (1) (1)

4 - 4

0

The discriminant is zero.

a = 1, b = -2, c = 2

b 2 - 4ac

(-2) 2 - 4 (1) (2)

4 - 8

-4

The discriminant is negative.

Graph of Related Function

Notice that the related function has

two x-intercepts.


one x-intercept.


no x-intercepts.

Number of Solutions

two real solutions one real solution no real solutions

If b 2 - 4ac > 0, the equation has two real solutions.

If b 2 - 4ac = 0, the equation has one real solution.

If b 2 - 4ac < 0, the equation has no real solutions.

The Discriminant of Quadratic Equation a x 2 + bx + c = 0

3E X A M P L E Using the Discriminant

Find the number of real solutions of each equation using the discriminant.

A 3 x 2 + 10x + 2 = 0a = 3, b = 10, c = 2

b 2 - 4ac 10 2 - 4 (3) (2) 100 - 24 76 b 2 - 4ac is positive.There are two realsolutions.

B 9 x 2 - 6x + 1 = 0a = 9, b = -6, c = 1

b 2 - 4ac (-6) 2 - 4 (9) (1) 36 - 36 0 b 2 - 4ac is zero.There is one real solution.

C x 2 + x + 1 = 0a = 1, b = 1, c = 1

b 2 - 4ac 1 2 - 4 (1) (1) 1 - 4 -3 b 2 - 4ac is negative.There are no real solutions.

Find the number of real solutions of each equation using the discriminant.

3a. 2 x 2 - 2x + 3 = 0 3b. x 2 + 4x + 4 = 0 3c. x 2 - 9x + 4 = 0

584 Chapter 8 Quadratic Functions and Equations



Example 3. Find the number of solutions of each equation using the discriminant.

A. ! B. ! C. ! 3x2 − 2x + 2 = 0 2x2 +11x +12 = 0 x2 + 8x +16 = 0


Guided Practice. Find the number of solutions of each equation using the discriminant.

10. ! 11. ! 12. !

8-9 The Quadratic Formula and the Discriminant (p 588-589) 31, 32, 33, 35-38.

2x2 − 2x + 3= 0 x2 + 4x + 4 = 0 x2 − 9x + 4 = 0


The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the beginning height of the object above the ground.

If the object is shot straight up from the ground, the initial height of the object above the ground equals 0.

Helpful Hint

20 ft

15 ft

10 ft

5 ft

The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = -16 t 2 + vt + c, where c is the beginning height of the object above the ground.

4E X A M P L E Physics Application

A weight 1 foot above the ground on a carnival strength test is shot straight up with an initial velocity of 35 feet per second. Will it ring the bell at the top of the pole? Use the discriminant to explain your answer.

h = -16 t 2 + vt + c20 = - 16t 2 + 35t + 1 Substitute 20 for h, 35 for v, and 1 for c.

Subtract 20 from both sides.

Evaluate the discriminant.

Substitute -16 for a, 35 for b, and -19 for c.

0 = - 16t 2 + 35t + (-19) b 2 - 4ac

35 2 - 4 (-16) (-19) = 9

The discriminant is positive, so the equation has two solutions. The weight will reach a height of 20 feet so it will ring the bell.

4. What if…? Suppose the weight is shot straight up with an initial velocity of 20 feet per second. Will it ring the bell? Use the discriminant to explain your answer.

There is no one correct way to solve a quadratic equation. Many quadratic equations can be solved using several different methods.

5E X A M P L E Solving Using Different Methods

Solve x 2 + 7x + 6 = 0.

Method 1 Solve by graphing.y = x 2 + 7x + 6 Write the related quadratic

function and graph it.

Factor.

Use the Zero Product Property.

Solve each equation.

Add ( b __ 2 ) 2 to both sides.

Factor and simplify.

Take the square root of both sides.


The solutions are the x-intercepts, -6 and -1.

Method 2 Solve by factoring.

x 2 + 7x + 6 = 0 (x + 6) (x + 1) = 0x + 6 = 0 or x - 1 = 0 x = -6 or x = -1

Method 3 Solve by completing the square.

x 2 + 7x + 6 = 0 x 2 + 7x = -6

x 2 + 7x + 49 _ 4

= -6 + 49 _ 4

(x + 7 _ 2

) 2 = 25 _

4

x + 7 _ 2

= ± 5 _ 2

x + 7 _ 2

= 5 _ 2

or x + 7 _ 2

= - 5 _ 2

x = -1 or x = -6





Example 4. The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the initial height of the object above the ground. The ringer on a carnival strength test is 2 feet off the ground and is shot upward with an initial velocity of 30 feet per second. Will it reach a height of 20 feet? Use the discriminant to explain your answer.


14. Guided Practice. Suppose the weight is shot straight up with an initial velocity of 20 feet per second from 1 foot above the ground. Will it ring the bell? Use the discriminant to explain your answer.



20 ft

15 ft

10 ft

5 ft

The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = -16 t 2 + vt + c, where c is the beginning height of the object above the ground.

4E X A M P L E Physics Application

A weight 1 foot above the ground on a carnival strength test is shot straight up with an initial velocity of 35 feet per second. Will it ring the bell at the top of the pole? Use the discriminant to explain your answer.

h = -16 t 2 + vt + c20 = - 16t 2 + 35t + 1 Substitute 20 for h, 35 for v, and 1 for c.

Subtract 20 from both sides.

Evaluate the discriminant.

Substitute -16 for a, 35 for b, and -19 for c.

0 = - 16t 2 + 35t + (-19) b 2 - 4ac

35 2 - 4 (-16) (-19) = 9

The discriminant is positive, so the equation has two solutions. The weight will reach a height of 20 feet so it will ring the bell.

4. What if…? Suppose the weight is shot straight up with an initial velocity of 20 feet per second. Will it ring the bell? Use the discriminant to explain your answer.


5E X A M P L E Solving Using Different Methods

Solve x 2 + 7x + 6 = 0.

Method 1 Solve by graphing.y = x 2 + 7x + 6 Write the related quadratic

function and graph it.

Factor.

Use the Zero Product Property.


Add ( b __ 2 ) 2 to both sides.

Factor and simplify.

Take the square root of both sides.


The solutions are the x-intercepts, -6 and -1.

Method 2 Solve by factoring.

x 2 + 7x + 6 = 0 (x + 6) (x + 1) = 0x + 6 = 0 or x - 1 = 0 x = -6 or x = -1

Method 3 Solve by completing the square.

x 2 + 7x + 6 = 0 x 2 + 7x = -6

x 2 + 7x + 49 _ 4

= -6 + 49 _ 4

(x + 7 _ 2

) 2 = 25 _

4

x + 7 _ 2

= ± 5 _ 2

x + 7 _ 2

= 5 _ 2

or x + 7 _ 2

= - 5 _ 2

x = -1 or x = -6




Method 4 Solve using the Quadratic Formula.

1 x 2 + 7x + 6 = 0 Identify a, b, and c.

Substitute 1 for a, 7 for b, and 6 for c.

Simplify.



x = -7 ±

√ """"" 7 2 - 4 (1) (6)

__ 2 (1)

x = -7 ± √ """ 49 - 24 __ 2

= -7 ± √ " 25 _

2 = -7 ± 5 _

2

x = -7 + 5 _ 2

or x = -7 - 5 _ 2

x = -1 or x = -6

Solve. 5a. x 2 + 7x + 10 = 0 5b. -14 + x 2 = 5x 5c. 2 x 2 + 4x - 21 = 0

Notice that all of the methods in Example 5 produce the same solutions, -1 and -6. The only method you cannot use to solve x 2 + 7x + 6 = 0 is using square roots. Sometimes one method is better for solving certain types of equations. The table below gives some advantages and disadvantages of the different methods.

METHOD ADVANTAGES DISADVANTAGESGraphing • Always works to give approximate

solutions

• Can quickly see the number of solutions

• Cannot always get an exact solution

Factoring • Good method to try first

• Straightforward if the equation is factorable

• Complicated if the equation is not easily factorable

• Not all quadratic equations are factorable.

Using square roots

• Quick when the equation has no x-term

• Cannot easily use when there is an x-term

Completing the square

• Always works • Sometimes involves difficult calculations

Using the Quadratic Formula

• Always works

• Can always find exact solutions

• Other methods may be easier or less time consuming.

Methods of Solving Quadratic Equations

No matter what method I use, I like to check my answers for reasonableness by graphing.

I used the Quadratic Formula to solve 2 x 2 - 7x - 10 = 0. I found that x ≈ -1.09 and x ≈ 4.59. Then I graphed y = 2x 2 - 7x - 10. The x-intercepts appeared to be close to -1 and 4.5, so I knew my solutions were reasonable.

Solving Quadratic Equations

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586 Chapter 8 Quadratic Functions and Equations



Example 5. Solve x2 – 9x + 20 = 0. Show your work.


Guided Practice. Solve each of the following. 15. x2 – 9x + 20 = 0 16. –14 + x2 = 5x


17. 2x2 + 4x – 21 = 0


8-9 The Quadratic Formula and the Discriminant • Desmos: The Discriminant of a Quadratic Function (FZ4N) • (p 588-589) 31, 32, 33, 35-39, 41, 42, 52, 54, 55, 56. • 8B Are You Ready pretest & posttests.

algebra 8 9 study guide the quadratic formula and...

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